Domination type parameters of Pell graphs

Arda Buğra Özer,Elif Saygı,Zulfukar Saygı

doi:10.26493/1855-3974.2637.f61

Abstract

Pell graphs are defined on certain ternary strings as special subgraphs of Fibonacci cubes of odd index. In this work the domination number, total domination number, 2-packing number, connected domination number, paired domination number, and signed domination number of Pell graphs are studied. Using integer linear programming, exact values and some estimates for these numbers of small Pell graphs are obtained. Furthermore, some theoretical bounds are obtained for the domination numbers and total domination numbers of Pell graphs.

Highlights

One of the basic models for interconnection networks is the n-dimensional hypercube graph Qn. It has 2n vertices, represented by all binary strings of length n, and two vertices in Qn are adjacent if they differ in exactly one coordinate
The n dimensional Fibonacci cube Γn is defined as the subgraph of Qn induced by the vertices whose string representations are Fibonacci strings
A set D ⊆ V is called a total dominating set of a graph G without isolated vertices, if every vertex in V is adjacent to some vertex in D and the total domination number γt(G) of G is defined as the minimum cardinality of a total dominating set of G

Summary

Introduction

One of the basic models for interconnection networks is the n-dimensional hypercube graph Qn. The n dimensional Fibonacci cube Γn is defined as the subgraph of Qn induced by the vertices whose string representations are Fibonacci strings. They were introduced by Hsu [10] as an alternative model for interconnection networks and extensively studied in the literature [13]. A set D ⊆ V is called a total dominating set of a graph G without isolated vertices, if every vertex in V is adjacent to some vertex in D and the total domination number γt(G) of G is defined as the minimum cardinality of a total dominating set of G. The domination type parameters of Fibonacci and Lucas cubes are first considered in [3, 17]. We studied some domination type parameters of Pell graphs

Preliminaries

Main results

Integer linear programming for domination numbers

Additional domination type parameters of small Pell graphs